# Definition:Kernel (Category Theory)

## Definition

Let $C$ be a category.

Let $f : A \to B$ be a morphism.

### Definition 1: for categories with initial objects

Let $C$ have an initial object $0$.

A kernel of $f$ is a morphism $\operatorname{ker}(f) \to A$ which is a pullback of the unique morphism $0 \to B$ via $f$ to $A$.

### Definition 2: for categories with zero objects

Let $C$ have an zero object $0$.

A kernel of $f$ is the equalizer of $f$ and the zero morphism $0 : A \to B$.